Question: Simplify; express your answer in exponential form. Assume $q\neq 0, r\neq 0$. $\dfrac{{(q^{4}r^{2})^{3}}}{{(q^{5}r^{4})^{5}}}$
Answer: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(q^{4}r^{2})^{3} = (q^{4})^{3}(r^{2})^{3}}$ On the left, we have ${q^{4}}$ to the exponent ${3}$ . Now ${4 \times 3 = 12}$ , so ${(q^{4})^{3} = q^{12}}$ Apply the ideas above to simplify the equation. $\dfrac{{(q^{4}r^{2})^{3}}}{{(q^{5}r^{4})^{5}}} = \dfrac{{q^{12}r^{6}}}{{q^{25}r^{20}}}$ Break up the equation by variable and simplify. $\dfrac{{q^{12}r^{6}}}{{q^{25}r^{20}}} = \dfrac{{q^{12}}}{{q^{25}}} \cdot \dfrac{{r^{6}}}{{r^{20}}} = q^{{12} - {25}} \cdot r^{{6} - {20}} = q^{-13}r^{-14}$